Tool CaT
stdout:
MAYBE
Problem:
cond(true(),x,y) -> cond(gr(x,y),x,add(x,y))
gr(0(),x) -> false()
gr(s(x),0()) -> true()
gr(s(x),s(y)) -> gr(x,y)
add(0(),x) -> x
add(s(x),y) -> s(add(x,y))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond(true(), x, y) -> cond(gr(x, y), x, add(x, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4()
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^3)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
gr^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {4}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
gr^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_1() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
gr^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 1 0] x1 + [3 3 3] x2 + [0]
[3 0 0] [3 3 3] [0]
[3 0 0] [3 3 3] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_5(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4() = [0]
[0]
[0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4()}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 2 0] x1 + [2]
[0 1 4] [0]
[0 0 0] [2]
add^#(x1, x2) = [1 3 1] x1 + [0 0 0] x2 + [0]
[3 2 2] [0 0 4] [0]
[0 2 2] [0 0 2] [2]
c_4() = [1]
[0]
[0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 2] [3]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4()
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[3 0] [3 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_3(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {4}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_2() = [1]
[0]
c_3(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 1] x1 + [3 3] x2 + [0]
[3 0] [3 3] [0]
c_4() = [0]
[0]
c_5(x1) = [1 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_5(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4() = [0]
[0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4()}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [4]
[0 1] [0]
add^#(x1, x2) = [1 3] x1 + [0 0] x2 + [4]
[2 2] [4 4] [0]
c_4() = [1]
[0]
c_5(x1) = [1 0] x1 + [3]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4()
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_3(x1) = [1] x1 + [2]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4()}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
add^#(x1, x2) = [6] x1 + [7] x2 + [0]
c_4() = [1]
c_5(x1) = [1] x1 + [7]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond(true(), x, y) -> cond(gr(x, y), x, add(x, y))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4(x)
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^2)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^3)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
[3 0 0] [3 0 0] [0]
[3 0 0] [3 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [2]
[0 0 2] [2]
[0 0 0] [0]
gr^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
[2 2 0] [0 2 0] [0]
[4 0 0] [0 2 0] [0]
c_3(x1) = [1 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
* Path {4}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
[2]
s(x1) = [1 4 2] x1 + [0]
[0 0 2] [0]
[0 0 1] [0]
gr^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
[2 2 2] [0 0 4] [0]
[2 2 2] [0 0 0] [0]
c_1() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
* Path {4}->{3}: YES(?,O(n^3))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
[0]
s(x1) = [1 3 2] x1 + [2]
[0 1 2] [0]
[0 0 1] [2]
gr^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 2] [0]
[0 1 0] [0 2 4] [0]
c_2() = [1]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 0 0] [6]
* Path {6}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 1 0] x1 + [3 3 3] x2 + [0]
[3 0 0] [3 3 3] [0]
[3 0 0] [3 3 3] [0]
c_4(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 2] x1 + [2]
[0 0 2] [3]
[0 0 1] [2]
add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
[0 2 1] [0 0 0] [2]
[4 0 2] [0 0 4] [0]
c_5(x1) = [1 0 0] x1 + [1]
[0 0 0] [2]
[2 2 0] [3]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
true() = [0]
[0]
[0]
gr(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
false() = [0]
[0]
[0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
c_0(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
gr^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_4(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [3 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4(x)}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {1},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[0]
[0]
s(x1) = [1 0 5] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
add^#(x1, x2) = [1 0 3] x1 + [2 0 0] x2 + [0]
[2 0 0] [0 4 0] [4]
[0 0 0] [0 2 0] [2]
c_4(x1) = [1 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 2] [0]
[0 0 0] [2]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4(x)
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^2)) ]
->{4} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^2)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^2))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[3 0] [3 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 1] [2]
gr^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
[0 2] [0 0] [0]
c_3(x1) = [1 2] x1 + [5]
[0 0] [3]
* Path {4}->{2}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [0]
gr^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
[4 1] [2 0] [0]
c_1() = [1]
[0]
c_3(x1) = [1 0] x1 + [3]
[0 0] [7]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [0]
[0]
s(x1) = [1 4] x1 + [2]
[0 0] [0]
gr^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_2() = [1]
[0]
c_3(x1) = [1 2] x1 + [3]
[0 0] [0]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[3 0] [3 3] [0]
c_4(x1) = [3 0] x1 + [0]
[0 0] [0]
c_5(x1) = [2 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 4] [4]
c_5(x1) = [1 0] x1 + [0]
[0 0] [3]
* Path {6}->{5}: YES(?,O(n^2))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
true() = [0]
[0]
gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
false() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [3 0] x1 + [0]
[0 0] [0]
gr^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3(x1) = [3 0] x1 + [0]
[0 0] [0]
add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_4(x1) = [1 1] x1 + [0]
[0 0] [0]
c_5(x1) = [3 0] x1 + [0]
[0 0] [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4(x)}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {1},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
s(x1) = [1 2] x1 + [2]
[0 1] [1]
add^#(x1, x2) = [3 2] x1 + [4 0] x2 + [0]
[2 5] [4 0] [0]
c_4(x1) = [3 0] x1 + [1]
[0 0] [0]
c_5(x1) = [1 0] x1 + [7]
[0 0] [7]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, 2: gr^#(0(), x) -> c_1()
, 3: gr^#(s(x), 0()) -> c_2()
, 4: gr^#(s(x), s(y)) -> c_3(gr^#(x, y))
, 5: add^#(0(), x) -> c_4(x)
, 6: add^#(s(x), y) -> c_5(add^#(x, y))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{6} [ YES(?,O(n^1)) ]
|
`->{5} [ YES(?,O(n^1)) ]
->{4} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
The weight gap principle does not apply:
The input cannot be shown compatible
Induced complexity for the usable rules: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ cond^#(true(), x, y) -> c_0(cond^#(gr(x, y), x, add(x, y)))
, gr(0(), x) -> false()
, gr(s(x), 0()) -> true()
, gr(s(x), s(y)) -> gr(x, y)
, add(0(), x) -> x
, add(s(x), y) -> s(add(x, y))}
Proof Output:
The input cannot be shown compatible
* Path {4}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [3] x1 + [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_3(x1) = [1] x1 + [7]
* Path {4}->{2}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [3] x1 + [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(0(), x) -> c_1()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_1) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [0] x2 + [4]
c_1() = [1]
c_3(x1) = [1] x1 + [2]
* Path {4}->{3}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [3] x1 + [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {gr^#(s(x), 0()) -> c_2()}
Weak Rules: {gr^#(s(x), s(y)) -> c_3(gr^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(gr^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [2]
gr^#(x1, x2) = [2] x1 + [2] x2 + [0]
c_2() = [1]
c_3(x1) = [1] x1 + [7]
* Path {6}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [1] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4(x1) = [3] x1 + [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_5(x1) = [1] x1 + [7]
* Path {6}->{5}: YES(?,O(n^1))
----------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following TMI:
The following argument positions are usable:
Uargs(cond) = {}, Uargs(true) = {}, Uargs(gr) = {},
Uargs(add) = {}, Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(gr^#) = {},
Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
true() = [0]
gr(x1, x2) = [0] x1 + [0] x2 + [0]
add(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
false() = [0]
s(x1) = [0] x1 + [0]
cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [3] x1 + [0]
gr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2() = [0]
c_3(x1) = [3] x1 + [0]
add^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [3] x1 + [0]
Induced complexity for the usable rules: YES(?,O(1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {add^#(0(), x) -> c_4(x)}
Weak Rules: {add^#(s(x), y) -> c_5(add^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(0) = {}, Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {1},
Uargs(c_5) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
s(x1) = [1] x1 + [0]
add^#(x1, x2) = [2] x1 + [7] x2 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.